Showing papers on "Distribution (differential geometry) published in 1946"

Geodesics on surfaces of revolution

Abstract: Introduction. We are concerned in this paper with the behavior in the large of the geodesic lines on a class of surfaces of revolution. The central theme is the number and the distribution of the double points of these geodesics, and one of the main theorems establishes a "zoning" of the surfaces in a manner dictated by this distribution. A second theorem sets up a classification of admissible surfaces on the basis of the number of the double points of its geodesic lines. An admissible surface S is formed by revolving about Oy a curve which rises monotonically from the origin to infinity as x increases, and which possesses a continuously turning tangent (save possibly at certain exceptional points). On every geodesic of S there is a point P, the "point of symmetry," which is nearest to the vertex of S, and at which the curve is tangent to a parallel of S. The two branches of the geodesic proceed in either direction from P, and spiral symmetrically in opposite senses around the axis of S toward infinity. Under special conditions the two branches may fail to intersect one another. More generally, however, they cut each other repeatedly. in a sequence of double points, which may conveniently be numbered by starting with the one nearest the vertex. We may thus speak of the "first double point," the "second double point;" and so on. This sequence may be finite or infinite, but if it is finite for one geodesic of S it is finite for all. The above discussion is of course not meant to apply to the meridians of S, whose special nature is perfectly clear. A discrete sequence of parallels, P1, P2, . . *, can be found on S dividing it into a corresponding sequence of "zones," Z1, Z2, * a a . The first zone Z is the portion of S containing the vertex and bounded by (but not including) the parallel P1. The nth zone Zn, for n > 1, is the portion of S bounded by Pn-i and Pn, including the points of the former parallel, but not those of the latter. In the case of a surface whose generating curve is tangent to Ox at 0, it is shown that every point of the nth zone is the 1st, 2nd, * * * , (n-1)th double point of certain geodesics of S, but is a double point of higher order of no geodesic of S. In the case of Z1 this is taken to mean that no point of this first zone, or "cap," is the double point of any geodesic of S. In the case of a generatrix not tangent to the x-axis this conclusion appears in a suitably modified form. The zones may be finite or infinite in number; and in particular the cap may extend over the whole of S.